Total Time Derivative Continuous Systems Homework for Ch. 4 and 5

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Total time derivative Continuous systems Homework for Ch. 4 and 5 Math Methods, D. Craig 2007–02–07 Total and partial time derivatives If you’re a stationary observer monitoring the temperature of the air as the wind goes by, and you detect an increase ∂T > 0, ∂t this is written with a partial because T = T(~r, t) In the definition of the partial derivative time is varied but other variables are fixed. If you’re in a ballon moving through stationary T field then ∂T/∂t = 0 while you can have dT > 0. dt This is because you are moving along at d~r/dt = ~v. 1 For the particular case of moving through a stationary T(~r): dT(~r) = (~v · ∇T) dt Problems 5.5 c–f work out the steps for get- ting the total time deriviative explicitly. In the general case of f = f(~r, t), then df(~r, t) ∂f(~r, t) = + ~v · ∇f(~r, t) dt ∂t is the total time derivative. It is related to the movement of a quantity. In a gas it may be the movement of a parcel of material carried around by the flow. 2 In general there are two ways to describe continuous systems (such as fluids): Eulerian description: describe every quantity at a fixed location in space. Then the change with properties in time is described by ∂d/∂t. Lagrangian description: follow a property as carried by the flow. Then change is described by total time derivative d/dt. Eulerian is often most convenient for numerical simulation, however if one must track particles or parcels, Lagrangian description is most use- ful. 3 Gradient in spherical coordinates ∂f 1 ∂f 1 ∂f ∇f = ^r + θ^ + φ^ ∂r r ∂θ r sin θ∂φ In cylindrical coordinates ∂f 1 ∂f ∂f ∇f = ^r + φ^ + z^ ∂r r ∂φ ∂z 4 Homework assignment 3 for Ch. 4 and 5 4.4c Taking the Jacobian with the determi- nant. 4.5 b and c Cylindrical coordinates. The geo- metric reasoning is easier than the spherical case. 5.4 f Extending force analysis to power. 5.5 g Showing the balloon and stationary ob- servers both work out with total time derivative. Due Tuesday Feb. 13 by 3 pm. 5.

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